Adsorption onto heterogeneous porous materials equilibria and kinetics

In chapter 3, the use of the Kierlik-Rosinberg density functional theory, for the calculation of multi-component sorption equilibria of simple Lennard-Jones molecules in a slit pore, is

Adsorption onto Heterogeneous Porous Materials: Equilibria and Kinetics

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr M Rem, voor een commissie aangewezen door het College voor Promoties in het openbaar te

verdedigen op dinsdag 5 juni 2001 om 16.00 uur

door

Frans Bernard Aarden

geboren te Standdaarbuiten

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr.ir P.J.A.M Kerkhof

en

prof.dr.ir K Kopinga

Copromotor:

dr.ir A.J.J van der Zanden

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Aarden, Frans B

Adsorption onto heterogeneous porous materials : equilibria and

kinetics / by Frans B Aarden - Eindhoven : Technische Universiteit Eindhoven, 2001

Voor mijn ouders

Summary

Adsorption technology is important for the separation and purification

of products, which often is the value adding step in process industry Together with the development of new materials, like zeolites and micro-porous membranes, this stresses the need to gain a better understanding in adsorption processes

The basic objective of the study in this thesis is to contribute to the understanding and modelling of the equilibria and kinetics of adsorption processes Special attention is given to the phase separation inside a porous material

In chapter 1, a short introduction to adsorption processes is presented

Chapter 2 discusses the properties of the materials that were used in this study Those materials were three types of activated carbon and several adsorbates The adsorbates were all organic compounds, namely, nitrobenzene, p-nitroaniline, quinoxaline, benzene and pyridine Nitrogen adsorption experiments and mercury porosimetry experiments were carried out to characterise the activated carbons The aqueous phase sorption isotherms of the organic adsorbates were measured at several temperatures The sorption isotherms were modelled with a Radke-Prausnitz isotherm This model fits the experimental data well over a large concentration range, compared to popular models such as the Langmuir, Freundlich and Jossens isotherms

In chapter 3, the use of the Kierlik-Rosinberg density functional theory, for the calculation of multi-component sorption equilibria of simple Lennard-Jones molecules in a slit pore, is discussed The model shows the existence of several pore filling mechanisms, dependent on molecular interactions, pore width and temperature These mechanisms are: continuous pore filling, film layer formation and capillary phase separation

The density functional theory also is used to calculate the pore size distribution

of the activated carbons, used in this study, from the nitrogen sorption isotherms It predicts pore sizes that are about twice as large than predicted by the Horvath-Kawazoe method, which is an often used method to characterise the pore size distribution of an activated carbon The prediction of the pore size may have important consequences in understanding kinetics of adsorption since the pores are only a few molecular diameter in width and the molecular motion

in the first molecular layer near the pore wall differs greatly from the kinetics of molecules, further away from the pore wall

In Chapter 4, a new model is put forward for the calculation of the intra-particle kinetics for adsorption from the liquid phase Contradictory to the usual models, which are based on surface diffusion, this one is based on the capillary flow of capillary phase separated adsorbate The model is applied to experimental data from literature on the adsorption of nitrobenzene and benzonitrile from the liquid phase onto activated carbon A good prediction is obtained for the

dependence of the effective diffusion coefficient on the amount adsorbed and temperature

Chapter 5 presents data on the kinetics of adsorption from water phase for nitrobenzene, p-nitroaniline and quinoxaline for the three activated carbons The experiments, that were carried out in order to obtain these data, were batch uptake experiments, in which the concentration in the aqueous solution was monitored during the adsorption process A surface diffusion theory, based on the Maxwell-Stefan equations, was used to model the experimental data Molecular diffusion in macro-pores and surface diffusion in micro-pores, were assumed to take place in parallel The Maxwell-Stefan surface diffusion coefficient was shown to have a large dependence, of several orders of magnitude, on the amount adsorbed One explanation for this is that at low concentrations, the adsorbate is strongly bound in small pores, near the pore walls, and has a small mobility At high concentrations, the adsorbate is also located in larger pores and farther away from the pore walls, resulting in a higher mobility Another explanation is that the transport by molecular diffusion

in macro-pores and the transport in micro-pores, that may be viewed as capillary transport or surface diffusion, take place in series The micro-pore transport is the faster mechanism and will become more dominant at higher carbon loading This results in a larger surface diffusion coefficient

In chapter 6, the fairly new application of magnetic resonance imaging (MRI) for the measurement of adsorption kinetics of nitrobenzene onto activated carbon is discussed MRI makes it possible to measure directly the concentration profiles of the adsorbate in the activated carbon Two experiments were carried out The first one is the adsorption from the vapour phase The second experiment is the adsorption from heavy water solution The results of the latter experiment show reasonable agreement with the results obtained from the experimental method of chapter 5 A big advantage of the MRI experiments

is that they are several orders of magnitude less time consuming than the experiments of chapter 5

Also, a T2-study of nitrobenzene on activated carbon was made with MRI Nitrobenzene shows three T2 peaks One of them can be attributed to nitrobenzene inside macro-pores The other two peaks are caused by nitrobenzene in micro-pores It is likely that one of those peaks originates from molecules in the first layer near the pore-wall and that the other one originates from molecules that are located farther away from the pore wall

To obtain better information about the adsorption process with MRI, it is desired to measure the adsorption profiles with a higher resolution and to measure the T2 data with less noise This should be possible by applying a stronger external magnetic field or by using a larger carbon sample

Samenvatting

Adsorptie is een belangrijke technologie in het scheiden en zuiveren van producten het is vaak de ‘value-adding’ step in de proces industrie Ook gezien de recente ontwikkelingen van nieuwe materialen, zoals zeolieten en micro-poreuze membranen, bestaat de behoefte om een beter inzicht te krijgen

in adsorptie processen

De basisdoelstelling van de studie in dit proefschrift is een bijdrage te leveren aan het begrijpen en modelleren van de evenwichten en kinetiek van adsorptie processen Speciale aandacht gaat uit naar fasenscheiding in poreuze materialen

In hoofdstuk 1 wordt een korte introductie betreffende adsorptie-processen gegeven

Hoofdstuk 2 bediscussieert de eigenschappen van de materialen die gebruikt werden in deze studie Dit waren 3 typen actieve kool en enkele adsorbaten De adsorbaten waren allen organisch, namelijk, nitrobenzeen, p-nitroaniline, quinoxaline, benzeen en pyridine Stikstof adsorptie-experimenten en kwikporosimetrie-experimenten werden uitgevoerd om de actieve kool te karakteriseren De waterfase sorptie-isothermen van de organische adsorbaten werden bij verschillende temperaturen gemeten Ze werden gemodelleerd met een Radke-Prausnitz isotherm Deze bleek de experimentele data over een groot gebied, vergeleken met populaire modellen als Langmuir, Freundlich en Jossens, goed te beschrijven

In hoofdstuk 3 wordt de Kierlik-Rosinberg-‘density functional theory’ gebruikt voor de berekening van multi-component sorptie-evenwichten van eenvoudige Lennard-Jones moleculen in een slit-porie Het model toont aan dat er verschillende mechanismen bestaan waarmee een porie gevuld kan worden Dit

is afhankelijk van moleculaire interacties, porie-grootte en de temperatuur De verschillende mechanismen zijn: continue porie-vulling, filmlaag-vorming en capillaire fasenscheiding

De ‘density functional theory’ werd ook gebruikt om de poriegrootte-verdeling van de verschillende koolsoorten te berekenen, uitgaande van de gemeten stikstof sorptie-isothermen De berekende poriediameters zijn ongeveer twee keer zo groot als die berekend met de Horvath-Kawazoe methode Horvath-Kawazoe is een veel gebruikte methode om poriegrootte-verdelingen van actieve kool te berekenen Het bepalen van de poriegrootte-verdeling kan belangrijke consequenties hebben voor het begrip van de adsorptie-kinetiek, omdat de poriën slechts enkele molecuuldiameters groot zijn De beweeglijkheid van moleculen direct tegen de poriewand wijkt namelijk aanzienlijk af van die van de moleculen die zich verder van de wand bevinden

In hoofdstuk 4 wordt een nieuw model geïntroduceerd voor de berekening van

de interne adsorptie-kinetiek voor adsorptie vanuit de waterfase In tegenstelling tot de gebruikelijke modellen, die gebaseerd zijn op oppervlakte-diffusie, is dit model gebaseerd op de capillaire stroming van capillair fasengescheiden adsorbaat Het model wordt toegepast op experimentele data uit de literatuur

van de adsorptie van nitrobenzeen en benzonitril vanuit de waterfase aan actieve kool Het model geeft een goede voorspelling van de temperatuur- en beladingsafhankelijkheid van de effectieve diffusiecoëfficiënt

Hoofdstuk 5 presenteert data betreffende de kinetiek van adsorptie vanuit de waterfase voor de adsorbaten nitrobenzeen, p-nitroaniline en quinoxaline voor

de drie verschillende koolsoorten De data werden verkregen via batch uptake experimenten, waarbij de concentratie in de waterfase werd gevolgd tijdens het adsorptie proces Een oppervlaktediffusie-theorie, gebaseerd op de Maxwell-Stefan oppervlaktediffusie-coëfficiënt, werd gebruikt om de experimentele data

te modelleren Moleculaire diffusie in de macro-poriën en diffusie in de poriën werden parallel verondersteld De dusdanig gevonden Maxwell-Stefan oppervlaktediffusie-coëfficiënt varieerde enkele ordegroottes met variërende geadsorbeerde hoeveelheid Een verklaring hiervoor is dat bij lage concentraties het adsorbaat sterk gebonden is in de kleine poriën dicht tegen de poriewand Dus heeft het een lage mobiliteit Bij hoge concentraties zit het adsorbaat ook in grotere poriën en verder verwijderd van de poriewand Dit resulteert in een hogere mobiliteit van de moleculen Een andere verklaring is dat het transport door moleculaire diffusie in macro-poriën en het transport in de micro-poriën in serie plaatsvinden Het micro-porie transport, dat gezien kan worden als oppervlakte-diffusie of als capillair transport, is het snellere mechanisme en wordt dominant bij hogere hoeveelheid geadsorbeerde stof Dit leidt tot een hogere oppervlakte-diffusiecoëfficiënt in het parallel-model

micro-In hoofdstuk 6 wordt de relatief nieuwe toepassing van MRI voor het meten van adsorptie-kinetiek besproken MRI werd gebruikt om direct de concentratie-profielen van nitrobenzeen in actieve kool te meten gedurende een adsorptie-experiment Er werden twee verschillende experimenten uitgevoerd Het eerste was de adsorptie van nitrobenzeendamp aan actieve kool Het tweede experiment was adsorptie vanuit een zwaar water oplossing De resultaten van het tweede experiment komen redelijk overeen met de resultaten die verkregen werden uit de batch uptake experimenten zoals beschreven in hoofdstuk 5 Een groot voordeel van de MRI experimenten is dat ze enkele ordegroottes minder tijdrovend zijn dan de batch uptake experimenten

Er werd ook een T2-analyse gemaakt van nitrobenzeen in actieve kool Nitrobenzeen geeft drie T2-pieken Eén van deze pieken kan worden toegeschreven aan nitrobenzeen dat zich in de macro-poriën bevindt De andere twee pieken worden veroorzaakt door nitrobenzeen in de micro-poriën Het is waarschijnlijk dat één van deze pieken afkomt van moleculen die zich direct tegen de poriewand bevinden De andere piek wordt dan veroorzaakt door moleculen die zich niet direct aan de poriewand bevinden

Om betere informatie over adsorptie-processen te verkrijgen met MRI experimenten, is het nodig om de adsorptie-profielen met een hogere resolutie

en de T2 data met relatief minder ruis meten Dit moet mogelijk zijn door het toepassen van een sterker extern magnetisch veld of door een groter monster te gebruiken

Contents

2.3.2 Characterisation of adsorbents with nitrogen adsorption 5

2.3.3 Characterisation of adsorbents with mercury porosimetry 9

3.2.1 The grand potential energy functional 22

3.2.2 Minimisation of the grand potential energy functional 24

3.2.3.3 Bulk phase separation, gas-liquid and liquid-liquid phase

equilibria 29

3.2.4 Adsorption of a Lennard-Jones fluid in a slit pore 30

3.3 Calculations and pore filling mechanisms 34

3.3.2 Nitrogen adsorption in a slit pore 40

3.3.2.3 Adsorption in an infinite slit pore 41

3.3.2.4 The use of DFT data to calculate pore size distributions from

sorption isotherms of the carbons used in this study 45

3.3.3 Multi-component adsorption in a carbon slit pore 50

4 Capillary transport in adsorption from liquid phase on activated carbon 64

4.2.1 Model I: Continuous capillary transport 68

4.2.2 Model II: Discontinuous capillary transport 70

5.3 Experiments in small concentration steps 80

5.4 Maxwell-Stefan diffusion model for transport in porous media 81

6 The application of 1 H-MRI to measurement of adsorption processes 94

6.1 Introduction to Magnetic Resonance Imaging (MRI) techniques 94

6.2 Measurement of nitrobenzene adsorption profiles with the aid of 1H-MRI 97

6.2.2.2 The MRI equipment and parameters 99

6.2.3.2 Nitrobenzene profiles as a function of time 101

6.2.4.1 Determination of the diffusion coefficient 105

6.3 Measurement of nitrobenzene adsorption profiles on activated carbon

6.3.3.1 Adsorption profiles from MRI measurements 109

6.3.3.2 Adsorption profiles from the batch uptake experiments 110

1 Introduction

1.1 Adsorption processes

The value-adding step in the process industry is often found in the separation and purification of products Adsorption technology has an important share in this step Also, adsorption is important in the removal of undesired components from, for example wastewater and air streams There are many materials, both from nature and from synthesis, that have sorption capacity The most important commercial adsorbents are activated carbon, molecular-sieve zeolites, silica gel and activated alumina The applications for these adsorbents depend on their particular sorptive properties [Crittenden and Thomas, 1998, p.28]

Applications of the use of activated carbon are, among other things: recovery of nitrogen from air, ethene from methane and hydrogen, removal of odours from gases, in gas masks and water purification, including removal of phenol, halogenated compounds, pesticides, caprolactam and chlorine

Zeolites are distinct from other adsorbents in that, for each type, there is no distribution of pore size because the crystal lattice into which the adsorbate molecules can or cannot enter is precisely uniform For this reason, zeolites are capable of separating effectively on the basis of molecular size They are used, among others for the separation of oxygen and argon, separation of normal from branched paraffins, drying of gases and removing water from azeotropic mixtures

Silica gel and activated alumina are used for the drying of gases, organic solvents and transformer oils

1.2 Objective of the study

The basic objective of the study in this thesis is to contribute to the understanding and modelling of the equilibria and kinetics of adsorption processes Special attention is given to phase separation phenomena inside a porous material

Sorption kinetics and equilibria were widely studied the past century This resulted in important developments as the Dubinin-Polanyi [Dubinin and Astakhov, 1971] potential theory In addition, a breakthrough was the recognition of the importance of considering the chemical potential gradient, rather than the concentration gradient, as the driving force for transport in adsorption systems [Habgood 1958; Krishna, 1993] This resulted in the wide-spread use of the Maxwell-Stefan equations in adsorption modelling

The development of special micro-porous membranes and zeolites has contributed to a more extensive use of adsorption processes in separation technology This also motivates the need for better understanding of adsorption principles

With the increase of computational power, detailed calculations can be performed on molecular scale The main methods are molecular dynamics,

Monte Carlo and density functional calculations These methods have led to insights, in the adsorption mechanism, that could not be obtained from experimental methods [Vlugt, 2000; Lastoskie et al., 1993]

On the experimental level, Magnetic Resonance Imaging is a promising tool that has not been applied extensively in studying adsorption processes yet It can be used to investigate sorption kinetics [Ruthven, 2000]

In this study, use was made of several of the developments in adsorption that were described above in order to gain understanding in adsorption processes

1.3 Outline of the thesis

The model material that is used in this study is activated carbon Even though its porous structure is very complicated, it was chosen because it is the most used adsorbent, world-wide

In chapter 2, the different kinds of activated carbon, used in the study, are characterised Liquid-phase sorption isotherms of several aromatic adsorbates were measured

In chapter 3, the Kierlik and Rosinberg [1991] density functional theory is used

to gain understanding in adsorption equilibria of one or two components in an infinite slit pore The influence of temperature and pore size was studied In addition, the effects of the interactions between the adsorbed component and the pore-wall and the interactions between components mutually were investigated

In chapter 4, a new model is put forward for calculation of the intra-particle kinetics for adsorption from the liquid phase Contradictory to the usual models, which are based on surface diffusion, this one is based on the capillary flow of capillary phase separated adsorbate

Chapter 5 treats the experimental kinetics of the adsorption from liquid phase for several activated carbons and aromatic adsorbates Batch uptake experiments were carried out in small concentration steps to determine the kinetics as a function of the amount adsorbed A surface diffusion theory, based

on the Maxwell-Stefan equations, was used to model the experimental data

In chapter 6, the fairly new application of magnetic resonance imaging for the measurement of adsorption kinetics is discussed The kinetics of nitrobenzene adsorption onto activated carbon were measured from both gas and liquid phase

In the latter case, a comparison was made with the conventional experimental method of chapter 5

References

Crittenden, Barry and Thomas, W John; ‘Adsorption technology and design, Butterworth-Heinemann’, Oxford, 1998

Dubinin, M.M and Astakhov, Y.A.; Adv Chem Ser., 102, pp 69-85, 1971

Habgood, H.W.; ‘The kinetics of molecular sieve action Sorption of nitrogen-methane mixtures by Linde molecular sieve 4A’, Can J Chem., 36, pp 1384-1397, 1958

Kierlik, E and Rosinberg, M.L.; ‘Density-functional theory for inhomogeneous fluids:

Adsorption of binary mixtures’, Physical review A, 44, no 8, pp 5025-5037, 1991

Krishna R.; ‘Problems and pitfalls in the use of the Fick formulation for intra-particle diffusion’, Chem Eng Sci, Vol 48, No 5, p.845, 1993

Lastoskie, C., Gubbins, K.E., and Quirke, N.; ‘Pore size heterogeneity and the carbon

slit pore: a density functional theory model’, Langmuir 9, pp 2693-2702, 1993

Ruthven, D.M., ‘Past progress and future challenges in adsorption research’, Ind Eng Chem Res, 39, pp 2127-2131, 2000

Vlugt, Thijs J.H.; ‘Adsorption and diffusion in zeolites: A computational study’, PhD Thesis, University of Amsterdam, 2000

2 Material properties

2.1 Introduction

In this section, the properties of the adsorbates and adsorbents, that were used in this study, are presented The adsorbents were characterised with nitrogen adsorption and mercury porosimetry experiments

The adsorbates that were used are benzene derivatives, having different properties with respect to a.o melting point and solubility in water

Benzene derivatives are common pollutants in potable and wastewater They can be effectively removed from water with the aid of activated carbon, which

is one of the most important types of adsorbent This is due to its low raw material costs, its large internal surface area and because it is non-hazardous Three types of activated carbon were used in this study The aqueous phase sorption isotherms of the adsorbates, in combination with the different activated carbons, were measured at several temperatures

2.2 Adsorbates

The adsorbates that were chosen for this study are quinoxaline (QNX), nitrobenzene (NBZ), p-nitroaniline (PNA), benzene (BZ) and pyridine (PYR) Their structural formulas are given below:

quinoxaline nitrobenzene p-nitroaniline benzene pyridine

Fig 2.2-1 Structural formulas of the used adsorbates

NBZ and PNA are often used in adsorption research, which makes it easy to obtain additional information on them NBZ is a liquid at room temperature PNA has a melting point of 149°C QNX has a melting point at 32°C This way

it is possible to perform experiments with QNX in an aqueous environment above and below the melting point In addition, NBZ and PNA are poorly soluble in water while QNX is very well soluble in water All atoms of the three molecules are almost positioned in the flat benzene plane For PNA and NBZ, it has been shown that they are reversibly adsorbed onto activated carbon [Tamon and Okazaki, 1996]

BZ is poorly soluble in water while PYR is completely miscible with water at room temperature Because their structural formulas are very similar, this makes

N

Some more detailed information on the adsorbates is given in table 2.2-1

Table 2.2-1 Characteristics of adsorbates

Molecular weight kg.mole -1 0.1301 0.1381 0.1231 0.0781 0.0791

D a,w is the bulk diffusivity of the adsorbate in water, from Kouyoumdjiev [1992, p 63]

C s is the solubility of the adsorbate in water

2.3 Adsorbents

2.3.1 Physical properties

Three types of activated carbon were used in this study All were in

extruded form and had a cylindrical shape The activated carbons RWB1 and

AP4-60 were manufactured for removal of organic pollutants from industrial as

well as municipal wastewater The AC R1-extra was manufactured for gas

separations

Physical properties of both carbons are given in table 2.3.1-1

Table 2.3.1-1 Physical properties of the activated carbons

1 determined from mercury porosimetry experiments

2 explanation follows in section 2.3.2

3 explanation follows in section 2.3.3

2.3.2 Characterisation of adsorbents with nitrogen adsorption

The carbons were characterised by performing nitrogen sorption measurements with a Coulter Omnisorp 100 The experiments were performed

at 77K, the boiling point of nitrogen at atmospheric pressure

The sorption isotherms are shown in figure 2.3.2-1 Here T is the absolute

temperature, R g is the gas constant [J.mole-1.K-1] and P is the nitrogen vapour

pressure [Pa] P 0 is the saturated nitrogen vapour pressure, which is atmospheric

at this temperature

Fig 2.3.2-1 N2 sorption isotherms at 77K for the activated carbons RWB1,

R1 extra and AP4-60

In view of the isotherms, it is observed that a significant amount of nitrogen was already adsorbed on the carbons RWB1 and R1-extra at very low pressures

(0.01 Pa) The hysteresis loop of all three isotherms closes at P/P 0=0.45, which

is common for nitrogen adsorption [Gregg and Sing, 1980, p.154] Below this pressure, the pore filling mechanism is so-called micro-pore filling, for which

no hysteresis occurs Micro-pores have a diameter of less than 2 nm The pore capacity of each carbon for nitrogen was read from the adsorption isotherm

micro-at P/P 0=0.45 These are tabulated in table 2.3.1-1 Note the large difference between the micro-pore capacities of AC RWB1 and AC R1 extra The micro-pore volume per volume of carbon particle was calculated by approximating the density of adsorbed nitrogen with its liquid bulk density at 77K (808 kg.m-3) The results are shown in table 2.3.1-1 When the nitrogen pressure exceeds

P/P 0=0.45 meso-pores are filled by means of capillary condensation In view of the figures, it is concluded that the meso-pore volume of the three carbons was quite small compared to the micro-pore volume

The micro-pore size distribution of each activated carbon was calculated with the Horvath and Kawazoe method [Horvath and Kawazoe, 1983] This approach was used to obtain the average potential energy for a slit shape pore from which

a method was derived to determine the micro-pore size distribution from the information of experimental isotherm data

The working equation [Do, 1998, p 319] runs as follows:

4 9

2 1 10

3 2 1

4 9

2 1 10

2 1 4

2 2 1 1 0

2

3 2

9

2

3 2

9 ln

σ σ

σ σ

σ σ

σ σ

σ σ

σ σ

σ σ σ

d d

d

A N A N N P

P

T

(2.3.2-1)

In equation 2.3.2-1, d is the diameter of the slit pore from pore wall centre to

pore wall centre N AV is the Avogadro number A pore with diameter d will be

filled with nitrogen when in equilibrium with a bulk phase pressure that is larger

than P/P 0

The parameters N 1 , N 2 , A 1 and A 2 were obtained from Do [1998, p 319] The

Lennard-Jones diameters of the adsorbent atom, carbon and of the adsorbate

atom, nitrogen are respectively σ1=0.34 nm and σ2=0.357 nm Thus, the

effective pore width is d-σ1

For

2 5

it is calculated that σ=0.299 nm Substituting all parameters in the working

equation, it follows that

10 3456 6 3485 0 2

10 664 2

697 0 2

44 44 ln

9

7 3

3 0

d d

d P

P

(2.3.2-3)

The micro-pore size distributions of the three activated carbons are shown in

figure 2.3.2-2

A more detailed analysis of the nitrogen isotherms is made in section 3.3.2 with

the aid of density functional theory

Fig 2.3.2-2 Cumulative amount of nitrogen adsorbed as a function of pore

size for the activated carbons RWB1, R1 extra and AP4-60, calculated with the Horvath Kawazoe method from the N 2

sorption isotherms at 77K

Fig 2.3.3-1 Cumulative pore size volumes for the activated carbons

RWB1, R1 extra and AP4-60, calculated with the Washburn equation from mercury intrusion experiments

RW B1

2.3.3 Characterisation of adsorbents with mercury porosimetry

The carbons were characterised by performing mercury porosimetry experiments with a Micromeritics Autopore IV 9500 apparatus The experiments were performed using the carbon particles in extruded form Cumulative pore size volumes were calculated from the intrusion experiments with the Washburn equation [Gregg and Sing, 1982, p 175] for cylindrical pores

The results are shown in figure 2.3.3-1

It becomes clear from figure 2.3.3-1 that AC R1-extra has a high macro-pore volume compared to the other two carbons The dimensions of most of the macro-pores vary roughly from 4 to 0.4 micrometer The macro-pore volume is calculated as the volume of all pores larger than 50 nm, which is the definition

of a macro-pore The values for the different carbons are given in table 2.3.1-1 Also from the mercury intrusion experiments, it is concluded that the meso-porous (pores with a diameter between 2 and 50 nm) volume is relatively small

After this time, the shaker was stopped, enabling the carbon particles to deposit This was done because the solution had to be filtered During filtering the temperature of the solution might change, which would influence the equilibrium Also, carbon particles in the solution silt up the filter, making filtering a slower process Thus the solution was decanted so that almost no carbon particles were left in it Then the solution was pressed through a, for the adsorbates inert, membrane filter (Schleicher and Schüll, diameter 47mm, pore diameter 0,45 µm)

The filtered solution was analysed with a Beckman DU-64 single-beam spectrophotometer with a wavelength range of 200 to 900nm at for the adsorbate characteristic minima and maxima in the light spectrum With the aid

of a calibration curve, the concentration was determined Because the minima in the spectrum of PNA coincide with the maxima in the spectrum of NBZ and the other way around, it was possible to determine the independent concentrations

of the components when both were present in the same aqueous solution This way it was possible to measure the multi-component sorption isotherm for this combination of components

2.4.2 Modelling of the isotherms

2.4.2.1 Radke-Prausnitz isotherm

The Radke-Prausnitz model [Radke and Prausnitz, 1972], was used to

fit the isotherm data The Radke-Prausnitz equation runs as follows:

RP

N RP RP

RP

C F K

C K

+

) / (

adsorbate concentration in the solvent K RP , F RP and N RP are the model

parameters, which are obtained by a non-linear statistical fit of the equation to

the experimental data

The Radke-Prausnitz equation has several important properties which make it

suitable for use in many adsorption systems At low concentrations it reduces to

a linear isotherm At high concentrations it becomes the Freundlich isotherm

and for the special case of N RP =0 it becomes the Langmuir isotherm This

Radke-Prausnitz model gives a good fit over a wide concentration range and

was therefore preferred above isotherm models like Langmuir, Freundlich and

Jossens

2.4.2.2 The potential theory

The potential theory is especially useful for adsorption of non-polar

components on micro-porous materials such as activated carbon The theory

interprets the sorption isotherm via a characteristic curve that is independent of

temperature This way it is possible to make predictions about the temperature

dependence of the sorption isotherm

Polanyi [Polanyi, 1932] proposed the concept of a pore wall surface force field

that can be represented by equi-potential contours above the surface, and that

the space between each set of equi-potential surfaces corresponds to a definite

adsorbed volume per kg of adsorbent, W, with

m

nV

Here, n is the number of moles adsorbed per unit mass of sorbent and V m the

molar volume of adsorbate in the adsorbed phase

As a consequence of the equi-potential contours, the cumulated volume of the

adsorbed space is a function of the adsorption potential, ε [J.mole-1] This is the

difference between the chemical potentials of the adsorbate, in the state of a

normal liquid, and the adsorbate, in the adsorbed state, at the same temperature

Then for low solute concentrations of the adsorbate

s

g C

C T

R ln

=

In which Cs [kg.m-3] is the adsorbate solubility

The characteristic curve is found by plotting nVm against

s g

C

C T

R ln According to the theory, sorption isotherms at different temperatures should

result in one characteristic curve

2.4.3 Results and discussion

The results of the measured single component isotherms are shown in figures 2.4.3-1 to 2.4.3-8 Also the Radke-Prausnitz fits are shown in the figures The fitting data are given in table 2.4.3-1 For PNA and NBZ, which have a low solubility in water, the isotherms at different temperatures were also plotted according to the potential theory The molar volumes were assumed to have negligible temperature dependence in the temperature range of 293K to

323K Thus the adsorbed volume W is directly proportional to carbon loading Q

Nitrobenzene seems to obey well to the potential theory in case of both the activated carbons RWB1 and R1-extra (figs 2.4.3-1b and 2.4.3-2b) However, for PNA, not one characteristic curve was found in the potential plot The curve

at 323K lies higher than that at 293K for both carbons If the temperature dependence of the molar volume had been taken into account, the difference would be even bigger, because the molar volume increases with temperature Figure 2.4.3-8 shows that benzene is much better adsorbed than pyridine on the activated carbon RWB1, from an aqueous solution This was to be expected since the benzene is less soluble in water than pyridine

The multi-component isotherm of the system PNA/NBZ on AC RWB1 at 293K

is shown in figure 2.4.3-9 Theories for adsorption equilibria in component systems are not as advanced as those for single component systems The slow progress in this area is due to a number of reasons [Do, 1998, p 248], among which: the lack of experimental data for multi-component systems and a solid surface that is too complex to model adequately

multi-Fig 2.4.3-1a Sorption isotherms of nitrobenzene on AC RWB1 The lines

are the Radke-Prausnitz fits Its parameters are given in table 2.4.3-1

Fig 2.4.3-1b Potential plot of sorption isotherms of nitrobenzene on AC RWB1

Fig 2.4.3-1c Sorption isotherms of nitrobenzene on AC RWB1 over wide

concentration range Fit 1 is Radke-Prausnitz fit of data points

in fig 2.3.4.1a For fit 2 all data points were taken into account

Fig 2.4.3-2a Sorption isotherms of nitrobenzene on AC R1 extra The lines

are the Radke-Prausnitz fits Its parameters are given in table 2.4.3-1

Fig 2.4.3-2b Potential plot of sorption isotherms of nitrobenzene on AC R1

Fig 2.4.3-3 Sorption isotherm of nitrobenzene on AC AP4-60 at 293K

The line is the Radke-Prausnitz fit Its parameters are given in table 2.4.3-1

Fig 2.4.3-4a Sorption isotherms of p-nitroaniline on AC RWB1 The lines

are the Radke-Prausnitz fits Its parameters are given in table 2.4.3-1

Fig 2.4.3-4b Potential plot of sorption isotherms of p-nitroaniline on AC

RWB1

Fig 2.4.3-5a Sorption isotherms of p-nitroaniline on AC R1-extra The

lines are the Radke-Prausnitz fits Its parameters are given in table 2.4.3-1

Fig 2.4.3-5b Potential plot of sorption isotherms of p-nitroaniline on AC

R1-extra

Fig 2.4.3-6 Sorption isotherms of quinoxaline on AC RWB1 The lines

are the Radke-Prausnitz fits Its parameters are given in table 2.4.3-1

ig 2.4.3-7 Sorption isotherms of quinoxaline on AC R1-extra The lines

are the Radke-Prausnitz fits Its parameters are given in table 2.4.3-1

Fig.2.4.3-8 Sorption isotherms of benzene and pyridine on AC RWB1 at

293K The lines are the Radke-Prausnitz fits Its parameters are given in table 2.4.3-1

Fig 2.4.3-9a Sorption isotherm of PNA on AC RWB1 at 293K as a

function of NBZ and PNA solute concentration (Data complemented with data from Kouyoumdiev [1992])

Fig 2.4.3-9b Sorption isotherm of NBZ on AC RWB1 at 293K as a

function of NBZ and PNA solute concentration (Data complemented with data from Kouyoumdiev [1992])

0.100.050.000.00.20.4

0.20.4

0.000.050.10

C NBZ [kg.

m-3 ]

Symbols

Greek symbols:

ε J.mole -1 adsorption potential

σ1 nm Lennard-Jones diameter of adsorbent atom

σ2 nm Lennard-Jones diameter of adsorbate atom

Latin symbols:

A1, A2, N 1 , N 2 parameters Horvath Kawazoe model

C kg.m -3 adsorbate concentration

D a,w m 2 s -1 bulk diffusivity of the adsorbate

F RP , K RP , N RP Radke-Prausnitz fit parameters

n mole.kgc-1 amount of adsorbate per kg carbon

P 0 Pa saturated vapour pressure

Q kg adsorbate kg c-1 amount of adsorbate per kg carbon

R g J.mole-1.K-1 gas constant

V m m 3 mole -1 molar volume

W m 3 kgc-1 adsorbed volume

3 Adsorption equilibria with density

functional theory

3.1 Introduction

In order to understand transport phenomena in heterogeneous porous materials, it is necessary to gain insight in the equilibrium behaviour in a single pore It has been known that fluids in narrow pores show a rich variety of phase transitions [Lastoskie et.al., 1993] It has also been known for a long time that the confinement leads to a shift of the bulk coexistence curve This is called capillary phase separation, when liquid-phase adsorption is considered, or capillary condensation, considering vapour-phase adsorption The past two decades statistical mechanics calculations made clear that in the micro-pores, with a diameter of a few molecular sizes, pore filling mechanisms like film-layer formation and continuous pore filling play an important role In addition, freezing and melting behaviour of confined fluids may deviate strongly from bulk behaviour [Sliwinska-Bartkowak et.al., 1999]

The numerical models of adsorption may be separated into three general categories Firstly the molecular dynamic (MD) simulations in which the forces between discrete molecules are calculated to follow their individual time-dependent movements Secondly, Monte Carlo (MC) methods replace the lengthy process of time integration with trial-and-error minimisation of the free energy followed by ensemble averaging Both the MD and MC simulations need the definition of discrete molecules that makes the simulations bound to a limited number of molecules in the calculations, because of computing time and memory limitations In contrast to MD and MC simulations, density functional theory (DFT) uses a mean field approximation for the molecule-molecule forces

so that the system is not limited to a certain number of molecules DFT accurately reproduces the results of MD and MC simulations with greatly reduced computational effort

Several versions of density functional theory have been reported in the literature All DFT’s use similar physical concepts Generally, it is assumed that the inter-molecular interaction can be divided into a short-range repulsive part, which determines the structure of the fluid, and a long-range attractive part The main difference between de different versions of DFT is the treatment of the short-range repulsive part, which is approached with a hard sphere fluid For highly inhomogeneous confined fluids, the local density is smoothed (non-local density approximation) The weighting function for smoothing the local density

is chosen to give a good description of the hard sphere direct pair correlation function for the uniform fluid over a wide range of densities

Here the Kierlik-Rosinberg DFT [Kierlik and Rosinberg, 1991] is used to gain insight in adsorption equilibria because it is applicable to multi-component fluids, in contrast to the often-used Tarazona model [Tarazona, 1985] It is numerically simple compared to other DFT’s like the Meister-Kroll-Groot theory [Meister and Kroll, 1985; Groot, 1987] or the Curtin-Ashcroft theory

[Curtin and Ashcroft, 1989], because the weighting functions are density

independent Finally, this theory also seems to be one of the most adequate ones

to describe adsorption of fluids at solid surfaces [Kierlik & Rosinberg, 1991,

Kozak & Sokolowski, 1991]

In the next section, the theory is treated for general pore geometry and simple

Lennard-Jones molecules Next, the theory is worked out further for slit-like

pore geometry In section 3.3 the calculations come up for discussion In section

3.3.1 it is explained how physical insight in the pore filling mechanisms is used

to find a numerical solution to the theory The subsequent section discusses

simulations of nitrogen adsorption onto activated carbon A comparison is made

with the experimental results of section 2.3.2 In section 3.3.3 the results of

some multi-component adsorption calculations are discussed

3.2 The Kierlik-Rosinberg DFT

3.2.1 The grand potential energy functional

The fluid density distribution in an area of uniform temperature, T, and

chemical potential, µi =µ∞,i, is determined by minimising the grand potential

energy functional [Kierlik and Rosinberg, 1990]

The intrinsic Helmholtz free energy functional F[{ρi }] describes the

contribution of the molecule-molecule interactions, which are modelled with a

12 , ,

r r

r

j j

LJ

σ σ

σi,,j [m] is the Lennard-Jones size parameter, εi,j [J] is the energy parameter

The attractive part of the Lennard-Jones potential is defined according to the

Weeks-Chandler-Andersen approximation [Weeks et.al.,1971] with a correction

for the potential at cut-off radius r c, beyond which the intermolecular

interactions are neglected The repulsive part of the Lennard-Jones potential is

approximated with the potential of an inhomogeneous hard sphere mixture

Thus, F is separated into an attractive term (F a ) and a repulsive term (F r)

1

r r r

r r

in which n is the number of components u i,j [J] is the fluid-fluid potential for

molecules i and j According to the Weeks-Chandler-Andersen approximation

) ( )

( )

( )

( )

is the radius at which the derivative of the Lennard-Jones potential equals zero

The repulsive forces between the molecules are approximated with a

hard-sphere potential:

The hard sphere diameter d hs is approximated with the Barker-Henderson

equation [Barker & Henderson, 1967]

) ( exp

T k

r u d

B

R i i

where

) ( )

( )

( )

According to Kierlik and Rosinberg the following approximation is used for the

Helmholtz free energy functional of a hard sphere mixture [Kierlik &Rosinberg,

1990, 1992] F r is split up into an ideal gas term F r,id and an excess part F r,ex

ex r id

i

i i i

B i

The free energy density,ΦPY , is taken from the Percus-Yevick compressibility

equation of state for a uniform hard-sphere mixture

2 3

3 2 3

2 1 3

i

n n

1

, 1

)

where ωk (r) are four independent weight functions wherein R hs,i is the hard

sphere radius of component i:

r r R

8

1 )

π

δ π

Here Θ(r) is the Heaviside function, δ(r) is the Dirac delta function and the

primes denote successive derivatives of the delta function

3.2.2 Minimisation of the grand potential energy functional

The equilibrium density profile was found by minimising the grand

potential energy functional, which means solving

simultaneously for all components j over the total volume area r

All parameters were made dimensionless according to

In the following (sections 3.2.2, 3.2.3 and 3.2.4), the asterisks are left out for the

dimensionless parameters All parameters, used in this section, are

; )

( )

ρ δρ

ρ

δ

d

y d

for an arbitrary function y(r) Here the object δΩ[{ρ i }]/δρj (r) depends on ρj and

r but not on ε or y(r)

The grand potential is split up in different parts that each are differentiated

{ }

[ ]

) (

) ( )

( ) ( )

(

1 1

, ,

r

r r r

r r

n i

i i n

i

i i a

ex r id r

j

i

d d

v F

F F

δρ

ρ µ ρ

δ δρ

Firstly, the differentiation of the part describing the influence of the external

potential is considered, using equations (3.2.1-1) and (3.2.2-2)

=

≠

=

r r r

r r r

r r

r r

d y v d

d y v

d v

d

j

j j n

j

i

i

i i

) ( ) (

)]

( ) ( )[

( )

( ) (

(

) ( ) (

r

r r r

j j

n

i

i i

v

d v

(3.2.2-5)

A similar solution is found for differentiation of the chemical potential term

) ( )

(

) (

r

r

j j

,

1 ln

ε ρ

ε

ε ρ ρ

d

y y

d T d

d

y dF

j j j

j i id

r

r r

r r

) ( ) (

) ( )

( ) (

1 ln

) (

Λ +

+

− +

ε ρ

ε ρ

r r r

r r

r

r r

r r

d y

y y

T

d y

y T

j j

j j

j j

r

d y

T d

y dF

j j j

ε ρ ρ

ε

(3.2.2-10) and

{ }

[ ]

) (

)]

( ), ( ), ( ), ( [ )

(

3 2 1 0 ,

r

r r r r r

PY j

ρ

The common chain rules for the derivatives of functions are used for

differentiation of functionals too Thus for (3.2.2-12) it is found from (3.2.1-16)

−

+

−

+ +

− +

n n

n

n n n n

n n n n

n

n n n n

T F

j j

j j

j

j j

, 3

3 2 2

3

, 2

2 2

2 3

, 3 2 1

3

, 2 1 2 , 1

3

, 3 0 3 ,

0

,

) 1 ( 24

' 2 )

1 ( 24

' 3

1

' 1

' '

1

' ) 1 ln(

'

)

(

π π

i j

n

) (

Finally, the derivative of the molecular attraction part of the grand potential

functional needs to be determined:

0 ,

0

) ' ( )) ( ) ( ))(

' ( ) ' ( ( ' 2

1

) ' ( ) ( )) ' ( ) ' ( ( ' 2

1

) ' ( ) ' ( )) ( ) ( ( ' 2

+ +

− +

+

− +

ε ρ

ε

ρ ε

ρ ε

ρ ε ρ

ε

ε ρ

ρ

d

u y y

d d

d

u y

d d

d

u y

d d d

y dF

j j

j

n

j i

j

i

a

r r r

r r

r r r

r r r r r

r r

r r r

r r

r r

(3.2.2-15)

− +

1

) ' ( ) ( ) ' ( ' 2

1

) ' ( ) ' ( ) ( ' 2

r r

r r r

r r r r r r

r r r

r r r

j j

j

n

j i i

j i

n

j k

k

k j k

u y y

d d

u y

d d

u y

d d

ρ ρ

j i j

u y

d d

u y

d d

1

,

) ' ( ) ( ) ' ( ' 2

1

) ' ( ) ' ( ) ( ' 2

1

r r r r r r

r r r

r r r

i

d

y dF

1

, 0

) ' ( ) ' ( ) ( '

;

r r r

r r

ε

ε ρ

ρ

ε

(3.2.2-18) Together with equation 3.2.2-2, it follows from equation 3.2.2-18 that

{ }

i

j i j

n

i

j i

j

i ex r j

j

v d u

F T

µ ρ

δρ

ρ δ

ρ

= +

− +

+ Λ

∑∫= ( ' ) ( ' ) ' ( )

) ( )]

r r

r r r

r

r

(3.2.2-20)

The second term on the left-hand side is worked out further in section 3.2.4 for

the geometry of an infinite slit pore

3.2.3 Bulk

3.2.3.1 The bulk chemical potential

Equation 3.2.2-20 is used to calculate the chemical potential of

component j in the bulk (µ∞,j) when the bulk densities of all components are

known The external potential equals zero and the bulk is assumed to have a

homogeneous density Then, from equation 3.2.2-20 follows that

{ }

j n

i i j j

j ex r j

1

2 , , ,

, , ,

The smoothed densities, used in the calculation of ΦPY according to equations

3.2.1-18 to 3.2.1-21, are simplified to [Kierlik & Rosinberg, 1991, 1992], which

makes it easy to calculate the second term in equation 3.2.1-1

R n

2 , ,

R n

3 , ,

3.2.3.2 The bulk pressure

The bulk pressure, P, is calculated with [Groot, 1987]

T N

ex r T

N

id r T

N

a T

F V

F V

F V

F V

ρ ρ

ρ ρ

V T

N

V F V

F V

F

,

2

, ,

) / (

ρ

ρ ρ

x i remains constant while the number of particles N remains constant

From equations 3.2.1-4 and 3.2.3-11 and 3.2.3-12 follows

j i j j

n i

n j

j j i T

N

a

dr r r u x x

d u

x x V

F

2 , , , 2

, , , 2

,

) ( 2

) ( 2

F

B T

,

ρ ρ

B T

3.2.3.3 Bulk phase separation, gas-liquid and liquid-liquid phase equilibria

At phase coexistence, the chemical potentials µi in the phases α and β

are equal for all components i:

The phase rule says:

degrees of freedom = number of components - number of phases + 2

When the liquid-vapour equilibrium for one pure component at fixed

temperature is calculated, there are two phases Thus, there are no degrees of

freedom left There are two unknowns, namely the liquid and the vapour

density Thus, to solve the problem, equation 3.2.3-16 is used, together with the

equation that says that the pressure in the liquid phase must equal that in the

vapour phase:

β

If the liquid-liquid equilibrium, for two components at fixed temperature, is

calculated, there are two phases and there is one degree of freedom left In this

case, the pressure is fixed too There are four unknowns Equation 3.2.3-16 is

used for both, i=1 and i=2, together with:

fixed

P P

Another way to calculate the liquid-liquid equilibrium is to use the vapour

pressure of the liquid as the total system pressure This means that there are 3

phases and thus that there is no degree of freedom left when temperature is

fixed

There are six unknowns, namely the densities of each of the two components in

the three phases The six equations that are used to solve the problem are

i i

3.2.4 Adsorption of a Lennard-Jones fluid in a slit pore

Consider the slit pore as illustrated in figure 3.2.4-1

Fig 3.2.4-1 Schematic representation of a slit-pore with pore-width w, and

molecular diameter σ

Also consider a function f x , which is only a function of the direction x,

perpendicular to the pore wall and a function f r, only a function of the distance

from point r Then the integration

f F

'

0

' ) ' ( ) ' ( )

(

r

r

r r r r

x x x f r rdrdx f

−

− +

+

= n

i

i hs x

i hs x

i hs

i hs x

i hs x

R x f R x f R

R x f R x f x

F

1

, ,

,

, ,

0

)]

( ' ) (

' [ 4

)]

( ) (

[ 2

1 )

x w z σ